Reduced complexity model responses

This model was derived from the following design spaces for each structural GDM type and should be considered valid within these settings. Note that projection beyond this region, increases the error and is not advised. Figure 53 shows the ‘design cube’ for the gO model created. The model is considered correct within the limits of the factors shown in the cube:

• 110µm > Thickness of GDM > 420µm. • 31.8% > porosity of GDM > 73%. • 60.3O

4-155 Figure 54 shows the ‘design cube’ for the Wmax model created. The model is considered correct with the limits of the factors shown in the cube:

• 110µm > Thickness of GDM > 420µm. • 31.8% > porosity of GDM > 73%. • 60.3O

C > Mean temperature during polarisation test > 73 OC.

Using the model outside of these limits increases the error in an undefined way and is not advised. The numeric values shown adjacent to each corner of the design cube (Figure 53, and Figure 54) are the model outputs, at the extreme values, for each of the displayed factors.

4-156 Figure 53: 'Design Cube' for gO models

4-157 Figure 54: 'Design Cube' for Wmax models

(non-woven (top), paper (middle) and woven (bottom) GDMs)

It is important to bear in mind that there are more than three factors in each model, but it is difficult to visualise this.

4-158 4.12. Reduced complexity regression for GDMs conclusion

The backwards step elimination of insignificant terms (detailed in sections 3.7, 3.8, 4.5, 4.6, 4.7 and 4.8), reduced the number of terms in the model (i.e. reduced complexity) without a reduction in the usefulness of the models created as shown in section 4.10. The models have been shown to be robust (i.e. not unduly influenced by extreme values) as Adjusted R2 = Predicted R2 +/- 0.2 [81,84]; this was shown in Table 22 and Table 33 for the gO and Wmax models respectively. Having established the credential of the proposed model the response surface plots generated were used to explore the design space. The response surface plots, shown (Figure 55 through Figure 60), map the developed models (gO model shown in Table 24 with categoric factor detail in Table 25, the Wmax model in Table 35 with categoric factor detail in Table 36, Table 37 and Table 38 ) across all points, within the limits of the design cube as discussed in section 4.11.1 (Figure 53 and Figure 54). The models graphically represent the result for mean temperature (Tbar) and porosity, and their impact on the model output results (either the gradient of the Ohmic region for gO plots or the peak power result for Wmax plots). The base plane on the response surface plots (Figure 55 through Figure 60) show a yellow field with contour lines to highlight lines of curvature in the response surface. In all cases (Figure 55 through Figure 60) the contour lines are straight, and no measurable curvature was detected. Findings discussed in the initial conclusions in section 4.6 (other than those attributed to machine variables) remain true. It was found that:

• In the gO plots (Figure 55, Figure 57 and Figure 59) the response for porosity dominates across all settings except for through-plane thickness.

• In the Wmax plots (Figure 56, Figure 58 and Figure 60) the response for porosity dominates across all settings except for through-plane thickness.

4-159 Figure 55: gO Through-thickness ‘Felt’ interactions

(Through-plane thickness 110 µm (top) and 420µm (bottom))

Examining Figure 55 in more detail, a comparison is made of the impact through-plane thickness of the GDM has on the performance of the Ohmic gradient for ‘felt’ GDMs. Overall there is a significant change, with lower thickness (110µm ‘top’) GDMs favouring lower porosity levels. That is to say, an Ohmic loss gradient closer to zero is considered preferable; as this indicates mass transfer losses are less likely to limit the performance of the fuel cell ( see section 1.1 and Figure 4 for further clarification). Conversely, thicker through-plane GDMs (420µm ‘bottom’), perform equally well across a wide range of porosity values, though fail to achieve the preferred gradient. It is interesting to note there is a ‘region of stability’ at approximately 53% porosity (highlighted with orange lines in Figure 55), where there is very little change to the gO result.

4-160 Figure 56: Wmax through-thickness ‘Felt’ interactions

(Through-plane thickness 110 µm (top) and 420µm (bottom))

Examining Figure 56 in more detail, a comparison is made of the impact through-plane thickness has on the performance of the peak power (in W.cm-2) for ‘felt’ GDMs. Overall, there is a significant change with lower thickness (110µm ‘top’) GDMs favouring lower porosity levels. That is to say, the maximum achievable peak power is 0.562 W.cm-2 (see section 1.1 and Figure 4 for further clarification). It is apparent that the lower thickness model performs less well at very high porosities, with the minimal peak power reading 0.158 W.cm-2. Conversely, thicker through-plane GDMs (420µm ‘bottom’) show a more uniform performance across a wide range of porosity values (maximum = 0.522 W.cm-2, minimum = 0.246 W.cm-2). It is interesting to note the ‘region of stability’ (highlighted with orange lines in Figure 55), is also evident, though for the Wmax model,

4-161 this region has shifted to a lesser value of porosity of approximately 42.5% (highlighted with black lines in Figure 56).

Figure 57: gO Through-thickness Paper interactions (Through-plane thickness 110 µm (top) and 420µm (bottom))

Examining Figure 57 in detail, a comparison is made of the impact through-plane thickness of the GDM has on the performance of the Ohmic gradient for ‘paper’ GDMs. Overall, there is a significant change with lower thickness (110µm ‘top’) GDMs favouring lower porosity levels. That is to say, an Ohmic loss gradient closer to zero is considered preferable; as this indicates mass transfer losses are less likely to limit the performance of the fuel cell (see section 1.1 and Figure 4 for further clarification). Conversely, thicker through-plane GDMs (420µm ‘bottom’) perform equally well across a wide range of porosity values, though fail to achieve the preferred gradient. It is interesting

4-162 to note, there is a ‘region of stability’ at approximately 53% porosity (highlighted with orange lines in Figure 57), where there is very little change to the gO result.

Figure 58: Wmax Through-thickness paper interactions (Through-plane thickness 110 µm (top) and 420µm (bottom))

In Figure 58, a comparison is made of through-plane thickness of the GDM on peak power (in W.cm-2) for ‘paper’ GDMs. Overall there is a significant change with lower thickness (110µm ‘top’) GDMs favouring lower porosity levels. That is to say the maximum achievable peak power is 0.592 W.cm-2 (see section 1.1 and Figure 4 for further clarification). It is apparent that the lower thickness model performs less well at very high porosities, with the minimal peak power reading 0.190 W.cm-2. Conversely, thicker through-plane GDMs (420µm ‘bottom’), shows a more uniform performance across a wide range of porosity values (maximum = 0.554 W.cm-2 , minimum = 0.276 W.cm-2 ). Note

4-163 a ‘region of stability’ (highlighted with orange lines in Figure 57), is also evident, though for the Wmax model, this region has shifted to a lesser value of porosity of approximately 42.5% porosity (highlighted with black lines in Figure 58). In the Wmax model for paper GDM, there is also a slight increase in peak power (+0.02 W.cm-2) at approximately 42.5% porosity point for thicker 420µm GDMs compared to the thinnest (110 µm) GDMs.

Figure 59: gO through-thickness woven interactions (Through-plane thickness 110 µm (top) and 420µm (bottom))

Figure 59 details a comparison of through-plane thickness of GDM, on the performance of the Ohmic gradient for ‘paper’ GDMs. Overall, there is a significant change with lower thickness (110µm ‘top’) GDMs favouring lower porosity levels. That is to say, an Ohmic loss gradient closer to zero is considered preferable; as this indicates mass transfer losses are less likely to limit the performance of

4-164 the fuel cell (see section 1.1 and Figure 4 for further clarification). Conversely, thicker through-plane GDMs (420µm ‘bottom’), perform equally well across a wide range of porosity values, though fail to achieve the preferred gradient. Note there is a ‘region of stability’ at approximately 52.5% porosity (highlighted with orange lines in Figure 59), where there is very little change to the gO result.

Figure 60: Wmax Through-thickness Woven interactions (Through-plane thickness 110 µm (top) and 420µm (bottom))

In Figure 60, a comparison is made of through-plane thickness of GDM on peak power 0.48 W.cm-2 woven GDMs. Overall, there is a significant change with lower thickness (110µm ‘top’) GDMs favouring lower porosity levels. That is to say, the maximum achievable peak power is 0.473 W.cm-2 (see section 1.1 and Figure 4 for further clarification). It is apparent that the lower thickness model

4-165 performs less well at very high porosities, with the minimal peak power reading 0.076 W.cm-2. Conversely, thicker through-plane GDMs (420µm ‘bottom’) show a more uniform performance across a wide range of porosity values (maximum = 0.436 W.cm-2, minimum = 0.165 W.cm-2). Note the ‘region of stability’ (highlighted with orange lines in Figure 59) is also evident, though for the Wmax model this region has shifted to a lesser value of porosity; approximately 42% porosity (highlighted with black lines in Figure 60).

The porosity and temperature surface response plots demonstrate a very strong through thickness effect. The peak performance for the Ohmic region (i.e. gO approaches zero) is achieved in all three structures tested, by minimising the through-plane thickness. The impact of porosity on the effect is harder to explain. It can seen that the lower porosity values (less than around 50%) are preferable, perhaps due the increased density of the system.

Even the inclusion of temperature does not significantly alter the effect, despite the well-established links to temperature and performance in every other metric measured. The minimisation of the porosity and through-plane thickness factors, dominates all other considerations. This remains true across all structure types, with woven structures performing moderately worse. Discussions around the impact of pores tend to focus on mass transport through the system. Larger pores require the least amount of pressure to allow water to penetrate [119]. The pore size and capillary pressure relationship is given as:

where γ = the surface energy of water, and rpore is the radius of a given pore and θ is the pore filling

factor [119] as detailed in section 4.2.1. The resultant pressure value, is the amount pressure required to force water to enter a pore of this size. This concept is the basis of the strong recommendation that a variety of pore sizes is preferential to optimise performance. Smaller pores will remain open for gas flow, while larger pores will dominate liquid water transport. Based on this theory, woven GDMs should dominate performance in most cases. The bimodal pore distribution between threads and the weft-warf of the weave itself, maximises the bimodal pore distribution in an ideal fashion [119]. Other authors have also commented on the importance of pore size and its distribution [120,121], and support this concept; with pore size distribution identified as being more important than either the mean pore size or total porosity in standard GDM materials. It should be observed at this point, that the construction of the ‘diffusion layer’ proposed in some papers referenced in this section, varies considerably from that used in these experiments; though the central arguments are still valid.

∆𝑃𝑃 =2𝛾𝛾𝑤𝑤𝑎𝑎𝐹𝐹𝑙𝑙𝑟𝑟_𝑟𝑟 𝐸𝐸𝐷𝐷𝑠𝑠𝜃𝜃

4-166 Jordan et al. (2000) [122] comment on the impact of the thickness of diffusion layers, and claim thinner GDMs are important for air based fuel cells (as opposed to oxygen based). However, their work is focused on a porous backing layer between carbon cloths and the catalyst layer, and their use of the term ‘thickness’ actually relates to the mass loading of this MPL. It may still provide some insight into the results from work in this thesis. Figure 55 through Figure 60 give a graphical representation of the relationship between performance, porosity and thickness of the carbon fibre based GDM for all three structures. The minimal porosity effect for improved performance is true in all materials, for both the gradient of the Ohmic loss region and the absolute peak power. Peak power settings were strongly temperature dependent but otherwise mirror the Ohmic gradient result.

It is interesting to note, the results shown from the peak power models (see Table 35 through Table 38) plotted in Figure 56, Figure 58 and Figure 60 indicate that, in high-demand and maximum-power applications, paper structures would be predicted to outperform the woven material by a significant degree. The non-woven ‘felt’ class of materials also outperforms the woven structure. This counteracts the usual assumption that the broad pore distribution of woven materials, are better suited to operation regions of the cell where high volumes of liquid water occur. It is possible the high demand state has not been held for long enough period, to trigger the volume of water generation required, to see the direct benefit of the woven pore-size distribution effect. The peak power measurement is an approximately 30-second ‘window’ of time from the 250 second duration of the polarisation curves that generated this data. Returning to the two-dimensional model from section 4.2.1, the perturbation factor was altered. Perturbation was set to a high value (0.35) to simulate the increased difficulty of liquid mass transport through reduced pores size, and reduced the overall hight of the calculated region to reflect the reduced thickness of the GDM (once again making the assumptions about initial starting temperature as per equation ( 4-5) and the temperature profile seen in Figure 20). The results for the increased perturbation value and reduced through-plane thickness, are plotted graphically in Figure 61 and Figure 62.

4-167 Figure 61: Reduced geometry GDM

(Steady state temperature (top) and O2 concentration (bottom))

x across half channel / half land

y ( thr ough t he G D L)